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Abundant Superintegrable Systems and Hessian Structures

John Armstrong, Andreas Vollmer

TL;DR

This work demonstrates that abundant non-degenerate second-order superintegrable systems on Riemannian manifolds of constant sectional curvature naturally induce Hessian structures via flat dual connections. It provides explicit Hessian potentials and coordinates for representative 2D and 3D systems, and shows a deep link between Hessian potentials and structure functions through $A = -\frac{1}{3}(\nabla^3\phi + 4\kappa\, g\otimes d\phi)$ with $\phi = -3\psi_-$. The results unify geometric (Hessian) and algebraic (Killing-tensor) viewpoints, revealing self-duality only for the harmonic oscillator in flat settings and highlighting a broader framework for interpreting superintegrable systems as Hessian geometries. This has potential implications for understanding dualities, coordinate systems, and integrability structures in geometric mechanics and mathematical physics.

Abstract

We show that a large class of non-degenerate second-order (maximally) superintegrable systems gives rise to Hessian structures, which admit natural (Hessian) coordinates adapted to the superintegrable system. In particular, abundant superintegrable systems on Riemannian manifolds of constant sectional curvature fall into this class. We explicitly compute the natural Hessian coordinates for examples of non-degenerate second-order superintegrable systems in dimensions two and three.

Abundant Superintegrable Systems and Hessian Structures

TL;DR

This work demonstrates that abundant non-degenerate second-order superintegrable systems on Riemannian manifolds of constant sectional curvature naturally induce Hessian structures via flat dual connections. It provides explicit Hessian potentials and coordinates for representative 2D and 3D systems, and shows a deep link between Hessian potentials and structure functions through with . The results unify geometric (Hessian) and algebraic (Killing-tensor) viewpoints, revealing self-duality only for the harmonic oscillator in flat settings and highlighting a broader framework for interpreting superintegrable systems as Hessian geometries. This has potential implications for understanding dualities, coordinate systems, and integrability structures in geometric mechanics and mathematical physics.

Abstract

We show that a large class of non-degenerate second-order (maximally) superintegrable systems gives rise to Hessian structures, which admit natural (Hessian) coordinates adapted to the superintegrable system. In particular, abundant superintegrable systems on Riemannian manifolds of constant sectional curvature fall into this class. We explicitly compute the natural Hessian coordinates for examples of non-degenerate second-order superintegrable systems in dimensions two and three.
Paper Structure (17 sections, 3 theorems, 106 equations)

This paper contains 17 sections, 3 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Non-degenerate systems and Hessian structures
  3. Hessian potentials and structure functions of abundant systems
  4. Hessian coordinates for non-degenerate systems in dimensions two
  5. The $2$-dimensional harmonic oscillator system (ho)
  6. The Smorodinski-Winternitz system (i)
  7. The Smorodinski-Winternitz system (ii)
  8. The 2-sphere (s2)
  9. Hessian coordinates for non-degenerate systems in dimensions three
  10. The Harmonic Oscillator (HO)
  11. The Smorodinski-Winternitz system (I)
  12. The Smorodinski-Winternitz system (IIa)
  13. The Smorodinski-Winternitz system (IIb)
  14. The 3-sphere (S3)
  15. Self-duality for the Hessian potentials of flat abundant systems
  16. ...and 2 more sections

Key Result

Theorem 2.1

Let $(M,g)$ be a conformally flat Riemannian manifold of dimension $n\geq3$, which we assume to be endowed with a non-degenerate system with structure tensor $T$. Furthermore, we assume that where $\mathring{R}_{ij}$ denotes the trace-free part of the Ricci tensor and $R$ the scalar curvature, and where $\Pi_\circ$ is the projection onto the trace-free part (with respect to $g$). Then the underly

Theorems & Definitions (9)

  • Remark 1.1
  • Definition 1.2
  • Theorem 2.1
  • proof
  • Example 3.1
  • Remark 3.2
  • Proposition 3.3
  • Proposition 3.5
  • proof